math identity?

Determine whether 4sin (x) cos (x) +tan^2 (x) +1 = 1/(cos^2(x))  + sin(4x) is an identity

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  • 4 months ago

    if this equation/identity is not satisfied by even one value of x , it is an equation  -

    ............................................

    Let x = 0

    4sin (x) cos (x) +tan^2 (x) +1 = 1/(cos^2(x)) + sin(4x)

    => 4 sin (0) cos (0) + tan²(0) + 1   =   1/cos²0  +  sin (4*0)

    => 0 + 0 + 1 = 1 + 0 ................ True

    .........................................................

    Let x = 30°

    4sin (x) cos (x) +tan^2 (x) +1 = 1/(cos^2(x)) + sin(4x)

    => 4 sin 30 * cos 30 +  tan²(30) + 1 = 1/cos²30 + sin (4*30)

    =>√3 + 1/3 + 1  ≠  4/3 + √3/2   .................... 

    Hence this is an equation.

  • 4 months ago

    It isn't!

    1/cos^2(x) = sec^2(x) = tan^2(x) + 1 ... one of the Pythagorean identities4sin(x)cos(x) = 2* 2sin(x)cos(x) = 2sin(2x)... sin(2x) = 2sin(x)cos(x) is a standard identity

    so sec^2(x) + 2sin(2x) = 4sin(x)cos(x) + tan^2(x) + 1

    would be the identity.

  • cosmo
    Lv 7
    4 months ago

    Not an identity.  It's only true for x = N pi/2, it's not true for any other value of x.   So, consider x = pi/4 it's a counter-example, the left side = 4 and the right side = 2.

    It's true to first order at N pi/2.

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